Two sides of a triangle are 6 in and 8 in. Using the Triangle Inequality Theorem (see picture above), which amount could be the length of the third side from the choices given?

Question

(1 point)
Responses

10 in.
10 in.

2 in.
2 in.

1 in.
1 in.

Answer 1

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides of lengths 6 in and 8 in, we can apply the theorem as follows:

1. The sum of the two sides must be greater than the third side:

   • \(6 + 8 > x\) → \(14 > x\) → \(x < 14\)

2. The difference of the two sides must be less than the third side:

   • \(6 + x > 8\) → \(x > 2\)
   • \(8 + x > 6\) → \(x > -2\) (This condition is always true since \(x\) has to be positive)

Combining these inequalities, we find that the length of the third side \(x\) must satisfy:

• \(2 < x < 14\)

Now, looking at the choices provided:

1. 10 in. - This satisfies the inequality \(2 < 10 < 14\).
2. 2 in. - This does not satisfy the inequality since \(x\) must be greater than 2.
3. 1 in. - This does not satisfy the inequality since \(x\) must be greater than 2.

Thus, the only valid option for the length of the third side is 10 in.